In electromagnetic equipment having a permanent magnet incorporated therein, it is indispensable to take account of a magnetic flux density (B) versus magnetic field (H) curve (hereinafter called a B-H curve) of a permanent magnet. The permanent magnet is naturally employed while being magnetized in any method.
A general behavior of a permanent magnet on a B-H curve and a way of handling the permanent magnet will be described below in case of their being needed in due course for description of the present invention.
FIG. 1 shows an example of a B-H curve of a permanent magnet. The horizontal axis indicates a magnetic field H, and the vertical axis indicates a magnetic flux density B. When the permanent magnet is not magnetized 1, if an external magnetic field is applied to the permanent magnet, the magnetic flux density B increases along an initial magnetization curve 2. At this time, as shown in FIG. 1, the magnetic flux density B initially sharply increases. Thereafter, a saturation phenomenon takes place as seen in a domain 3.
When the applied magnetic field H is diminished in the above state, the magnetic flux density B traces a curve that specifies larger values than the values specified by the initial magnetization curve 2. When the applied magnetic field H becomes null, the magnetic flux density B takes on a value 4 larger than zero. The value 4 represents a remanent magnetic flux density. The permanent magnet exhibits a large value of the remanent magnetic flux density. In case the permanent magnet is made of neodymium (Nd) that is a rare earth element, the remanent magnetic flux density reaches approximately 1.2 tesla (T).
Thereafter, when the magnetic field H is inversely applied, the magnetic flux density B moderately decreases. Thereafter, the magnetic flux density B sharply decreases to take on a negative value. At this time, an intercept 5 on the H axis represents a coercive force (Hc). In order to explicitly signify that the vertical axis indicates the magnetic flux density B, the coercive force is given a symbol bHc. In electromagnetic equipment employing a permanent magnet, a portion of a B-H curve from a point indicating a remanent magnetic flux density to a point indicating the coercive force is especially important. The portion is referred to as a demagnetization curve 6. As shown in FIG. 1, the magnetic flux density B changes along with a change in the applied magnetic field H.
The curves expressing a change in the magnetic flux density B with respect to the applied magnetic field H refer to a hysteresis loop. When the hysteresis loop is such that the amplitude of the applied magnetic field H is sufficiently large and the magnetic flux density B is saturated in both positive and negative domains, the hysteresis loop is called a major loop. Otherwise, the hysteresis loop is called a minor loop. A description has been made using terminology that is normally employed in physics and engineering. Even when any other terminology is employed, a person with an ordinary skill in the art can easily understand the description. The same applies to a description of the present invention to be made later.
As mentioned above, a B-H curve of a permanent magnet continuously changes over the first, second, third, and fourth quadrants in a magnetic field H versus magnetic flux density B orthogonal coordinate system. Therefore, a position of the permanent magnet on the B-H curve lies in any of the first, second, third, and fourth quadrants. A change in the position should fundamentally be continuously expressed whether it involves the quadrants or not.
By the way, as an index generally employed in designing a magnetic circuit that includes a permanent magnet, a permeance coefficient is available. The permeance coefficient will be briefed below.
A magnetic field which, as shown in FIG. 2, a coil 7 (N denotes the number of turns and I denotes the current) induces in an ambient space will be discussed below. In a magnetic flux tube 8 shown in the drawing, a line integral of a magnetic field H along a magnetic flux from a point A 9 to a point B 10 is provided, using an equation=−·φm,  (1)as∫ABHdl=−∫ABdφm=φm,A−φm,B=FAB.  (2)
A difference in a magnetic (scalar) potential φm is referred to as a magnetomotive force. If a path of integration encircles a current, the magnetomotive force comes to a sum of values of a current passing through a closed curve. When a magnetic flux density on a cross section of the magnetic flux tube 8 can be regarded as being constant, a magnetic flux flowing at an arbitrary position in the magnetic flux tube 8 is written, using an equationΦ=BS=μHS,  (3)as
                              F          AB                =                  Φ          ⁢                                    ∫              A              B                        ⁢                                                            ⅆ                  l                                                  μ                  ⁢                                                                          ⁢                  S                                            .                                                          (        4        )            
At this time, a magnetic resistance (reluctance) between the point A 9 and point B 10 is provided as
                                          R            AB                    =                                    ∫              A              B                        ⁢                                          ⅆ                l                                            μ                ⁢                                                                  ⁢                S                                                    ,                            (        5        )            and an equationFAB=ΦRAB  (6)is obtained.
A reciprocal of the reluctance or magnetic resistance is a permeance. Thus, when a general magnetic circuit is treated, the permeance is employed.
Next, a magnetic circuit including a permanent magnet will be discussed below. For brevity's sake, a description will be made of a case where a leakage magnetic flux is absent. Noted will be a magnetic flux density in an air gap 11 in a magnetic circuit shown in FIG. 3. Using subscripts g, i, and p, quantities relating to the air gap 11, an iron 12, and a permanent magnet 13 will be discriminated from one another. Assuming that H denotes a magnetic field in the permanent magnet 13, B denotes a magnetic flux density, l denotes a magnetic path length of each part, and S denotes a sectional area, an equation is drawn out due to the Ampere's rule asHglg+Hili+Hlp=0.  (7)
Since a magnetic flux remains constant on any section, a relationshipBSp=μ0HgSg=μiHiSi  (8)is established.
Herein, the Hi and Si values of the iron 12 are mean values obtained along the entire magnetic path. Accordingly, a permeance coefficient
                    p        =                              -                          B              H                                =                                                    l                p                                            S                p                                      ⁢                                          (                                                                            l                      g                                                                                      μ                        0                                            ⁢                                              S                        g                                                                              +                                                            l                      1                                                                                      μ                        l                                            ⁢                                              S                        l                                                                                            )                                            -                1                                                                        (        9        )            is obtained.
FIG. 4 shows a relationship of the magnetic flux density B and magnetic field H of the permanent magnet to a demagnetization curve. The vertical axis indicates the magnetic flux density B and the horizontal axis indicates the magnetic field H. The vertical axis and the horizontal axis intersect at an origin of coordinates 14. A demagnetization curve 15 crosses the vertical axis at a point of a remanent magnetic flux density 16, and crosses the horizontal axis at a point of a coercive force (bHc) 17. A point of a mean magnetic flux density of the permanent magnet and a point of a mean magnetic field thereof are present on the demagnetization curve 15. In FIG. 4, they shall coincide with an operating point 18. At this time, θ shall denote an angle 19 at which a half line extending from the origin of coordinates 14 to the operating point 18 meets the horizontal axis. Based on the definition of a permeance coefficient p, an equation
                    p        =                              -                          B              H                                =                                                                      l                  p                                                  S                  p                                            ⁢                                                (                                                                                    l                        g                                                                                              μ                          0                                                ⁢                                                  S                          g                                                                                      +                                                                  l                        i                                                                                              μ                          i                                                ⁢                                                  S                          i                                                                                                      )                                                  -                  1                                                      =                          tan              ⁡                              (                θ                )                                                                        (        10        )            is drawn out. The B and H values are determined with a position of the permanent magnet 13 on the demagnetization curve 15, that is, the operating point 18. A magnetic flux density in the air gap 11 is determined asBg=μ0Hg.  (11)
Meanwhile, an equation
                    R        =                              1            P                    =                                                    l                g                                                              μ                  0                                ⁢                                  S                  g                                                      +                                          l                i                                                              μ                  i                                ⁢                                  S                  i                                                                                        (        12        )            expresses a magnetic resistance of an external magnetic circuit viewed from the permanent magnet 13. Accordingly, an equation
                    p        =                  P          ⁢                                    l              p                                      S              p                                                          (        13        )            is drawn out. Therefore, the permeance coefficient p can be regarded as a value per unit volume of the permanent magnet 13 into which a permeance P of the external magnetic circuit is converted.
The foregoing way of thinking of a magnetic circuit has been adopted in the past. Before numerical analysis such as an analysis of the finite element method to be implemented by a computer is put to practical use, the way of thinking has widely been adopted as a design method for electromagnetic equipment employing a permanent magnet. As mentioned above, since it is easy to learn a physical relationship to a permeance P of an external magnetic circuit, when an operating point on a demagnetization curve of the permanent magnet is expressed, a permeance coefficient p has been generally used. A method for expressing the operating point with the permeance coefficient is widely adopted even at present when the numerical analysis such as the analysis of the finite element method to be implemented by a computer is used in practice. Examples are found in patent documents 1 and 2 (JP-A-2002-328956 and JP-A-2004-127056 respectively).
As described previously, a method for expressing an operating point of a permanent magnet using a permeance coefficient is quite advantageous because of the easiness in establishing a physical relationship to a magnetic circuit including a permanent magnet. Therefore, the method has been widely and generally used to date.
As described previously, the permeance coefficient has a physical meaning that is a value per unit volume of a permanent magnet into which a permeance P of an external magnetic circuit is converted. Since the permeance P of the magnetic circuit is a reciprocal of a magnetic resistance of the magnetic circuit and the magnetic resistance takes on a positive value, the permeance coefficient expressed by the equation (10) is also physically defined to take on a positive value. Namely, the permeance coefficient takes on a positive value determined with a geometrical shape.
As seen from the equation (10), when the operating point shifts from the first quadrant of a B-H curve to the second quadrant thereof, the permeance coefficient becomes discontinuous after exhibiting an infinite divergence (when the angle 19 (θ) shown in FIG. 4 is 90°). Therefore, as shown in FIG. 1, a position on the B-H curve of a permeance magnet, that is, an operating point lies in any of the first, second, third, and fourth quadrants. A change in the position should fundamentally be continuously expressed whether it involves the quadrants or not. Nevertheless, the change cannot be continuously expressed using the permeance coefficient.
Further, expressing an operating point of a permeance magnet using a permeance coefficient poses a problem in terms of design of electromagnetic equipment.
Needless to say, it is a magnetic flux density B and a magnetic field H that determines an operating point of a permeance magnet. As for the magnetic flux density B and magnetic field H, when they are indicated linearly, it is often more helpful in design of a magnetic circuit. More particularly, when a sectional area of a certain part of the magnetic circuit is increased 10%, the magnetic flux density B in the part is thought to decrease approximately 10%. In reality, even in a brochure of a permanent magnet manufacturer, a demagnetization curve is linearly plotted with respect to the magnetic flux density B and magnetic field H.
Even if a magnetic circuit is modified in order to decrease the magnetic flux density of a permanent magnet by 10%, a permeance coefficient representing an operating point of the permanent magnet on the demagnetization curve shown in FIG. 4 does not decrease 10%. This is because the permeance coefficient representing the operating point is defined with a slope as it is expressed by the equation (10). Specifically, electromagnetically, when the magnetic flux density B and magnetic field H are linearly indicated, a magnetic-circuit issue can be easily dealt with. Nevertheless, the permeance coefficient representing the operating point of the permanent magnet does not linearly change along with changes in the magnetic flux density B and magnetic field H respectively. This poses a problem in that it is hard to grasp the relationship among the magnetic flux density B and magnetic field H of the magnetic circuit and the permeance coefficient representing the operating point of the permanent magnet.
In addition, when the operating point of a permanent magnet is represented by a permeance coefficient, a problem described below further arises. The problem as well as the aforesaid problems will be detailed below.
A B-H curve of a permanent magnet fundamentally exhibits a nonlinear characteristic like the one shown in FIG. 1. However, the demagnetization curve 6 in FIG. 1 is often, as shown in FIG. 5, plotted in a simplified manner. In FIG. 5, the demagnetization curve 6 in FIG. 1 is plotted to include demagnetization curves 20 and 21. The demagnetization curve 20 intersects the vertical axis at a point of a remanent magnetic flux density 22. Operating points 23 and 24 are operating points in the domains of the demagnetization curves 20 and 21 respectively. A point of intersection between the demagnetization curve 20 and demagnetization curve 21 is a knickpoint 25 corresponding to a point on the demagnetization curve 6 in FIG. 1 at which the magnetic flux density sharply decreases. The knickpoint 25 may be called an irreversible demagnetization beginning point or a bend point. When the demagnetization curve can be, as shown in FIG. 5, treated approximately, the demagnetization curve is said to have excellent squareness. A permanent magnet made of Nd that is a rare earth element and has been adopted in recent years is said to be excellent in the squareness.
When an operating point lies in the domain of the demagnetization curve 20, whether a magnetic field H increases or decreases, the operating point 23 is thought to reversibly shift on the demagnetization curve 20. Therefore, a phenomenon that the permanent magnet irreversibly weakens due to the magnetic field H, that is, demagnetization is not thought to occur.
In contrast, when a negative magnetic field H (a magnetic field H in a negative direction on a B-H curve) increases, the operating point shifts to a domain on a negative magnetic field H side beyond the knickpoint 25, for example, to the operating point 24. In this case, even if the negative magnetic field H decreases to be null, the operating point does not return along the demagnetization curve 20 but traces a minor loop 26 lying below and substantially in parallel to the demagnetization curve 20. A remanent magnetic flux density becomes a remanent magnetic flux density 27 smaller than the initial remanent magnetic flux density 22. This is a demagnetization phenomenon that a magnet gets weakened. When demagnetization occurs, electromagnetic equipment employing the permanent magnet fails to exert the performance as initially designed. Therefore, it is quite significant that the permanent magnet is designed so as not to be demagnetized even when the electromagnetic equipment is operated.
As described later, in reality, an operating point exhibits a distribution inside a permanent magnet. Therefore, it is essential to accurately learn not only an average operating point of the permanent magnet but also a ratio at which certain part of the magnet indicated with a certain operating point occupies the whole of the magnet. When limit design is performed in terms of temperature or a magnetic field, if electromagnetic equipment including a permanent magnet is operated, it becomes necessary to accurately grasp a radio at which an internal part of the magnet that is demagnetized occupies the whole of the magnet. However, when the operating point is represented by a permeance coefficient, that is, expressed with a slope in the second quadrant of a B-H curve linearly plotted with respect to a magnetic flux density B and a magnetic field H, the relationship of the permeance coefficient to an associated area in the permanent magnet is hard to understand.
FIG. 6 shows an example in which a permeance coefficient representing an operating point is indicated in a graph in which a demagnetization curve and the operating point are plotted. The permeance coefficient takes on a value obtained by multiplying a value 28 in the drawing by a permeability μo in a vacuum. This kind of drawing is often carried in a brochure published from a magnet manufacturer. In this example, the permeance coefficient representing an operating point 29 is 1.2, and the permeance coefficient representing a knickpoint 30 is 0.4.
As apparent from FIG. 6, when a magnitude of a change in a negative magnetic field H is small, a magnitude of a change in the permeance coefficient is small in a domain in which the permeance coefficient takes on a small value. The magnitude of the change in the permeance coefficient is large in a domain in which the permeance coefficient takes on a large value. Therefore, in the domain in which the permeance coefficient takes on a large value, the change in the permeance coefficient looks large for a change in the negative magnetic field H. This poses a problem in that the relationship among the magnetic flux density B and magnetic field H of a magnetic circuit and the permeance coefficient representing an operating point of a permanent magnet is hard to grasp.